Finite Group Databasebeta

← #8735

Subgroups — #8735

Order 256.

257 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 257 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2274C211yesmaximal, characteristic
2C2×#324C212yesmaximal
3#1282C214yesmaximal
4C2×#117C411yescharacteristic
5C2×#117C411yescharacteristic
6C22×C4⋊C4C2211yescharacteristic
7C2×C22⋊C8C2214yes—
8C4×C4⋊C4—22no—
9C4×C4⋊C4—22no—
10C4×C22⋊C4C412yes—
11C4×C22⋊C4C2212yes—
12#324—22no—
13#324—22no—
14C23×C4C2×C411yescharacteristic
15C23×C4C2×C411yescharacteristic
16C23×C4C2311yescharacteristic
17C22×C8—24no—
18#117—22no—
19#117—24no—
20#117C814yescomplemented
21C2×C4⋊C4—21no—
22C2×C4⋊C4—21no—
23C2×C4⋊C4—24no—
24C2×C4⋊C4D412yes—
25C2×C4⋊C4D412yes—
26C2×C22⋊C4C2×C412yes—
27C2×C22⋊C4C2×C412yes—
28C2×C42—21no—
29C2×C42—21no—
30C2×C42—22no—
31C2×C42—22no—
32C22⋊C8—24no—
33C22⋊C8—24no—
34#95—22no—
35#95—22no—
36C24C22×C411yescharacteristic
37C22×C4—22no—
38C22×C4—22no—
39C22×C4—22no—
40C22×C4—22no—
41C22×C4—22no—
42C22×C4—81no—
43C22×C4—81no—
44C22×C4C22⋊C411yescharacteristic
45C22×C4C22⋊C411yescharacteristic
46C22×C4C22⋊C412yes—
47C22×C4C2×C812yes—
48C22×C4M1612yes—
49C22×C4C2×D412yes—
50C22×C4#4114yes—
51C2×C8—44no—
52C2×C8—84no—
53C4⋊C4—24no—
54C4⋊C4—24no—
55C4⋊C4—24no—
56C4⋊C4—24no—
57C22⋊C4—22no—
58C22⋊C4—22no—
59C22⋊C4C2×C814yes—
60C22⋊C4M1614yes—
61C42—24no—
62C42—24no—
63C42—82no—
64C42—82no—
65C23—41no—
66C23—41no—
67C23C2×C22⋊C411yescharacteristic
68C23#11712yes—
69C23C22×C811yescharacteristic
70C23C2×M1611yescharacteristic
71C23C2×M1612yes—
72C2×C4—21no—
73C2×C4—21no—
74C2×C4—21no—
75C2×C4—21no—
76C2×C4—24no—
77C2×C4—24no—
78C2×C4—22no—
79C2×C4—24no—
80C2×C4—24no—
81C2×C4—42no—
82C2×C4—44no—
83C2×C4—42no—
84C2×C4—42no—
85C2×C4—42no—
86C2×C4—42no—
87C2×C4—42no—
88C2×C4—82no—
89C2×C4C22⋊C812yes—
90C2×C4C22⋊C812yes—
91C2×C4#12314yes—
92C8—84no—
93C22—21no—
94C22—22no—
95C22—21no—
96C22—22no—
97C22—41no—
98C22—41no—
99C22—42no—
100C22#38111yescharacteristic
101C22C2×C22⋊C811yescharacteristic
102C22#40711yescharacteristic
103C22#43112yes—
104C22#43212yes—
105C4—22no—
106C4—22no—
107C4—24no—
108C4—44no—
109C4—44no—
110C4—42no—
111C2—21no—
112C2—21no—
113C2—22no—
114C2—22no—
115C2#181711yescharacteristic
116C2#182811yescharacteristic
117C2#183011yescharacteristic
subgroupG/H
center, socleC22#381
commutatorC23C22×C8
Frattini, cosocleC23×C4C23
prime radicalC24C22×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
20. #117C8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#22742
C2×#3242
#12822